Almost all k-colorable graphs are easy to color
Journal of Algorithms
The solution of some random NP-hard problems in polynomial expected time
Journal of Algorithms
A guided tour of Chernoff bounds
Information Processing Letters
A spectral technique for coloring random 3-colorable graphs (preliminary version)
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Coloring random and semi-random k-colorable graphs
Journal of Algorithms
Algorithms for coloring semi-random graphs
Random Structures & Algorithms
Computers and Intractability; A Guide to the Theory of NP-Completeness
Computers and Intractability; A Guide to the Theory of NP-Completeness
Improved Algorithms for Coloring Random Graphs
ISAAC '94 Proceedings of the 5th International Symposium on Algorithms and Computation
Coloring Semi-Random Graphs in Polynomial Expected Time
Proceedings of the 14th Conference on Foundations of Software Technology and Theoretical Computer Science
Zero Knowledge and the Chromatic Number
CCC '96 Proceedings of the 11th Annual IEEE Conference on Computational Complexity
Minimum coloring random and semi-random graphs in polynomial expected time
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Coloring Sparse Random Graphs in Polynominal Average Time
ESA '00 Proceedings of the 8th Annual European Symposium on Algorithms
Coloring sparse random k-colorable graphs in polynomial expected time
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
| Hi-index | 0.00 |
The k-colouring problem is to colour a given k-colourable graph with k colours. This problem is known to be NP-hard even for fixed k ≥ 3. The best known polynomial time approximation algorithms require nδ (for a positive constant δ depending on k) colours to colour an arbitrary k-colourable n-vertex graph. The situation is entirely different if we look at the average performance of an algorithm rather than its worst-case performance. It is well known that a k-colourable graph drawn from certain classes of distributions can be k-coloured almost surely in polynomial time.In this paper, we present further results in this direction. We consider k-colourable graphs drawn from the random model in which each allowed edge is chosen independently with probability p(n) after initially partitioning the vertex set into k colour classes. We present polynomial time algorithms of two different types. The first type of algorithm always runs in polynomial time and succeeds almost surely. Algorithms of this type have been proposed before, but our algorithms have provably exponentially small failure probabilities. The second type of algorithm always succeeds and has polynomial running time on average. Such algorithms are more useful and more difficult to obtain than the first type of algorithms. Our algorithms work as long as p(n) ≥ n−1+ε where ε is a constant greater than 1/4.